The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lov\'asz (1974) and Mader (1978) showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of $k$-hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Kir\'aly and Lau (Journal of Combinatorial Theory, 2008; FOCS 2006)). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.

翻译：无向图中的分裂操作是一种基本约简操作，它切断给定顶点关联的所有边，并在该顶点的邻点之间添加新边同时保持其度数。Lovász (1974) 和 Mader (1978) 分别证明了在特定条件下，该操作能在图中保持全局连通性和局部连通性。这些结果在图算法文献中具有深远应用。本文引入超图中的分裂操作。我们证明超图中存在一个保持局部连通性的完全分裂操作，并给出一个强多项式时间算法用于加权超图。通过展示两个应用说明超图中分裂操作的有效性：(1) 给出$k$-超边连通超图的构造性刻画；(2) 给出图与超图的最大斯坦纳根定向近似极小-极大关系的替代证明（源自 Király 与 Lau (Journal of Combinatorial Theory, 2008; FOCS 2006)）。我们对图近似极小-极大关系的证明绕过了 Nash-Williams 强定向定理，并采用了为超图开发的工具。